Abstract : A procedure is given for the computation of the fuel-optimal control for linear state-constrained systems where convex polygonal (possibly time-varying) sets are used for specifying the allowable control vectors and the allowable state vectors. The terminal condition can be imposed through a convex target set or through exact specification of the terminal state. The solution is obtained by formulating the problem so that the techniques of linear programming with upper bounds can be used. The use of these techniques and the special structure of the reformulated problem results in a computationally efficient algorithm. As an example, the algorithm is used for the solution of a space rendezvous problem. (Author)